If the lines joining the foci of the ellipse (x2/a2) +(y2/b2)=1, where ab, and an extremly of its minor axis are inclined at an angle 60° , then the eccentricity of the ellipse is

Question

If the lines joining the foci of the ellipse (x2/a2) +(y2/b2)=1, where a>b, and an extremly of its minor axis are inclined at an angle 60° , then the eccentricity of the ellipse is (A)  -(√3/2)  (B)  (1/2)  (C)  (√5/2)  (D)  (√7/3)

Tardigrade Admin · Accepted Answer

In Δ CBF, tan 30° =(FC/b) FC=(b/√3) ⇒ ae=(b/√3) ⇒ a2e2=(1/2)[a2(1-e2)] ⇒ 4e2=1 ⇒ e=(1/2)

Q. If the lines joining the foci of the ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1,$ where $a > b,$ and an extremly of its minor axis are inclined at an angle $60^{\circ},$ then the eccentricity of the ellipse is

$- \frac{3}{2}$

$\frac{1}{2}$

$\frac{5}{2}$

$\frac{7}{3}$

$3$

In $Δ CBF, tan 30^{\circ} = \frac{FC}{b}$
$FC = \frac{b}{3}$
$\Rightarrow$ $a e = \frac{b}{3}$
$\Rightarrow$ $a^{2} e^{2} = \frac{1}{2} [a^{2} (1 - e^{2})]$
$\Rightarrow$ $4 e^{2} = 1$
$\Rightarrow$ $e = \frac{1}{2}$

Q. If the lines joining the foci of the ellipse a2x2​+b2y2​=1, where a>b, and an extremly of its minor axis are inclined at an angle 60∘, then the eccentricity of the ellipse is

−23​​

21​

25​​

37​​

3​

In ΔCBF,tan30∘=bFC​ FC=3​b​ ⇒ ae=3​b​ ⇒ a2e2=21​[a2(1−e2)] ⇒ 4e2=1 ⇒ e=21​